Langberg, Michael and Sprintson, Alexander and Bruck, Jehoshua (2004) The Encoding Complexity of Network Coding. California Institute of Technology , Pasadena, CA. (Unpublished) http://resolver.caltech.edu/CaltechPARADISE:2004.ETR063

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Abstract
In the multicast network coding problem, a source s needs to deliver h packets to a set of k terminals over an underlying network G. The nodes of the coding network can be broadly categorized into two groups. The first group includes encoding nodes, i.e., nodes that generate new packets by combining data received from two or more incoming links. The second group includes forwarding nodes that can only duplicate and forward the incoming packets. Encoding nodes are, in general, more expensive due to the need to equip them with encoding capabilities. In addition, encoding nodes incur delay and increase the overall complexity of the network. Accordingly, in this paper we study the design of multicast coding networks with a limited number of encoding nodes. We prove that in an acyclic coding network, the number of encoding nodes required to achieve the capacity of the network is bounded by h^3k^2. Namely, we present (efficiently constructible) network codes that achieve capacity in which the total number of encoding nodes is independent of the size of the network and is bounded by h^3k^2. We show that the number of encoding nodes may depend both on h and k as we present acyclic instances of the multicast network coding problem in which [Omega](h^2k) encoding nodes are needed. In the general case of coding networks with cycles, we show that the number of encoding nodes is limited by the size of the feedback link set, i.e., the minimum number of links that must be removed from the network in order to eliminate cycles. Specifically, we prove that the number of encoding nodes is bounded by (2B+1)h^3k^2, where B is the minimum size of the feedback link set. Finally, we observe that determining or even crudely approximating the minimum number of encoding nodes needed to achieve the capacity for a given instance of the network coding problem is NPhard.
Item Type:  Report or Paper (Technical Report) 

Additional Information:  We would like to thank Matthew Cook for useful discussions. http://www.paradise.caltech.edu/papers/etr063.pdf 
Group:  Parallel and Distributed Systems Group 
Record Number:  CaltechPARADISE:2004.ETR063 
Persistent URL:  http://resolver.caltech.edu/CaltechPARADISE:2004.ETR063 
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ID Code:  26095 
Collection:  CaltechPARADISE 
Deposited By:  Imported from CaltechPARADISE 
Deposited On:  22 Mar 2005 
Last Modified:  26 Dec 2012 13:53 
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